Dirichlet series
| L(s) = 1 | + 96·4-s − 432·5-s + 240·7-s − 378·11-s + 1.68e3·13-s + 3.07e3·16-s − 2.82e3·19-s − 4.14e4·20-s + 7.62e4·23-s + 5.04e4·25-s + 2.30e4·28-s − 9.70e4·29-s + 2.14e4·31-s − 1.03e5·35-s − 2.55e4·37-s + 4.10e5·41-s + 7.14e4·43-s − 3.62e4·44-s − 3.47e5·47-s + 3.13e5·49-s + 1.61e5·52-s + 1.63e5·55-s − 3.69e5·59-s + 1.35e5·61-s − 6.55e4·64-s − 7.25e5·65-s − 2.89e5·67-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 3.45·5-s + 0.699·7-s − 0.283·11-s + 0.764·13-s + 3/4·16-s − 0.411·19-s − 5.18·20-s + 6.26·23-s + 3.23·25-s + 1.04·28-s − 3.98·29-s + 0.721·31-s − 2.41·35-s − 0.504·37-s + 5.95·41-s + 0.898·43-s − 0.425·44-s − 3.34·47-s + 2.66·49-s + 1.14·52-s + 0.981·55-s − 1.80·59-s + 0.598·61-s − 1/4·64-s − 2.64·65-s − 0.964·67-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{36}\right)^{s/2} \, \Gamma_{\C}(s+3)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 3^{36}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.35109\times 10^{13}\) |
| Root analytic conductor: | \(3.52461\) |
| Motivic weight: | \(6\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 3^{36} ,\ ( \ : [3]^{12} ),\ 1 )\) |
Particular Values
| \(L(\frac{7}{2})\) | \(\approx\) | \(0.02722765437\) |
| \(L(\frac12)\) | \(\approx\) | \(0.02722765437\) |
| \(L(4)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p^{5} T^{2} + p^{10} T^{4} )^{3} \) |
| 3 | \( 1 \) | |
| good | 5 | \( 1 + 432 T + 27228 p T^{2} + 31938624 T^{3} + 6557782944 T^{4} + 247791088872 p T^{5} + 8629838464196 p^{2} T^{6} + 283078864086552 p^{3} T^{7} + 1722605027432904 p^{5} T^{8} + 249266298149061408 p^{5} T^{9} + 6917392908490493484 p^{6} T^{10} + \)\(18\!\cdots\!24\)\( p^{7} T^{11} + \)\(47\!\cdots\!06\)\( p^{8} T^{12} + \)\(18\!\cdots\!24\)\( p^{13} T^{13} + 6917392908490493484 p^{18} T^{14} + 249266298149061408 p^{23} T^{15} + 1722605027432904 p^{29} T^{16} + 283078864086552 p^{33} T^{17} + 8629838464196 p^{38} T^{18} + 247791088872 p^{43} T^{19} + 6557782944 p^{48} T^{20} + 31938624 p^{54} T^{21} + 27228 p^{61} T^{22} + 432 p^{66} T^{23} + p^{72} T^{24} \) |
| 7 | \( 1 - 240 T - 256170 T^{2} + 96760144 T^{3} + 28756288698 T^{4} - 21652992911724 T^{5} - 28828710564488 T^{6} + 3581741463326135460 T^{7} - \)\(74\!\cdots\!06\)\( T^{8} - \)\(38\!\cdots\!96\)\( T^{9} + \)\(17\!\cdots\!86\)\( T^{10} + \)\(18\!\cdots\!44\)\( T^{11} - \)\(25\!\cdots\!18\)\( T^{12} + \)\(18\!\cdots\!44\)\( p^{6} T^{13} + \)\(17\!\cdots\!86\)\( p^{12} T^{14} - \)\(38\!\cdots\!96\)\( p^{18} T^{15} - \)\(74\!\cdots\!06\)\( p^{24} T^{16} + 3581741463326135460 p^{30} T^{17} - 28828710564488 p^{36} T^{18} - 21652992911724 p^{42} T^{19} + 28756288698 p^{48} T^{20} + 96760144 p^{54} T^{21} - 256170 p^{60} T^{22} - 240 p^{66} T^{23} + p^{72} T^{24} \) | |
| 11 | \( 1 + 378 T + 5413101 T^{2} + 2028148794 T^{3} + 18756094128375 T^{4} + 9398413208910060 T^{5} + 55739541025385760596 T^{6} + \)\(29\!\cdots\!76\)\( T^{7} + \)\(13\!\cdots\!97\)\( T^{8} + \)\(72\!\cdots\!30\)\( T^{9} + \)\(27\!\cdots\!65\)\( p T^{10} + \)\(14\!\cdots\!66\)\( T^{11} + \)\(56\!\cdots\!62\)\( T^{12} + \)\(14\!\cdots\!66\)\( p^{6} T^{13} + \)\(27\!\cdots\!65\)\( p^{13} T^{14} + \)\(72\!\cdots\!30\)\( p^{18} T^{15} + \)\(13\!\cdots\!97\)\( p^{24} T^{16} + \)\(29\!\cdots\!76\)\( p^{30} T^{17} + 55739541025385760596 p^{36} T^{18} + 9398413208910060 p^{42} T^{19} + 18756094128375 p^{48} T^{20} + 2028148794 p^{54} T^{21} + 5413101 p^{60} T^{22} + 378 p^{66} T^{23} + p^{72} T^{24} \) | |
| 13 | \( 1 - 1680 T - 17978772 T^{2} + 1125415024 p T^{3} + 195031695689304 T^{4} - 31662953227277808 T^{5} - \)\(14\!\cdots\!88\)\( T^{6} - \)\(36\!\cdots\!80\)\( T^{7} + \)\(84\!\cdots\!08\)\( T^{8} + \)\(30\!\cdots\!76\)\( T^{9} - \)\(41\!\cdots\!40\)\( T^{10} - \)\(60\!\cdots\!72\)\( p T^{11} + \)\(19\!\cdots\!26\)\( T^{12} - \)\(60\!\cdots\!72\)\( p^{7} T^{13} - \)\(41\!\cdots\!40\)\( p^{12} T^{14} + \)\(30\!\cdots\!76\)\( p^{18} T^{15} + \)\(84\!\cdots\!08\)\( p^{24} T^{16} - \)\(36\!\cdots\!80\)\( p^{30} T^{17} - \)\(14\!\cdots\!88\)\( p^{36} T^{18} - 31662953227277808 p^{42} T^{19} + 195031695689304 p^{48} T^{20} + 1125415024 p^{55} T^{21} - 17978772 p^{60} T^{22} - 1680 p^{66} T^{23} + p^{72} T^{24} \) | |
| 17 | \( 1 - 74302878 T^{2} + 4522701425194623 T^{4} - \)\(18\!\cdots\!74\)\( T^{6} + \)\(70\!\cdots\!67\)\( T^{8} - \)\(20\!\cdots\!92\)\( T^{10} + \)\(55\!\cdots\!70\)\( T^{12} - \)\(20\!\cdots\!92\)\( p^{12} T^{14} + \)\(70\!\cdots\!67\)\( p^{24} T^{16} - \)\(18\!\cdots\!74\)\( p^{36} T^{18} + 4522701425194623 p^{48} T^{20} - 74302878 p^{60} T^{22} + p^{72} T^{24} \) | |
| 19 | \( ( 1 + 1410 T + 90408279 T^{2} - 50956681126 T^{3} + 5240084810315283 T^{4} - 12275594334729075852 T^{5} + \)\(12\!\cdots\!10\)\( p T^{6} - 12275594334729075852 p^{6} T^{7} + 5240084810315283 p^{12} T^{8} - 50956681126 p^{18} T^{9} + 90408279 p^{24} T^{10} + 1410 p^{30} T^{11} + p^{36} T^{12} )^{2} \) | |
| 23 | \( 1 - 76248 T + 3008908950 T^{2} - 81660828897936 T^{3} + 1667774835599771802 T^{4} - \)\(26\!\cdots\!44\)\( T^{5} + \)\(34\!\cdots\!32\)\( T^{6} - \)\(37\!\cdots\!16\)\( T^{7} + \)\(41\!\cdots\!02\)\( T^{8} - \)\(58\!\cdots\!04\)\( T^{9} + \)\(10\!\cdots\!10\)\( T^{10} - \)\(16\!\cdots\!12\)\( T^{11} + \)\(21\!\cdots\!22\)\( T^{12} - \)\(16\!\cdots\!12\)\( p^{6} T^{13} + \)\(10\!\cdots\!10\)\( p^{12} T^{14} - \)\(58\!\cdots\!04\)\( p^{18} T^{15} + \)\(41\!\cdots\!02\)\( p^{24} T^{16} - \)\(37\!\cdots\!16\)\( p^{30} T^{17} + \)\(34\!\cdots\!32\)\( p^{36} T^{18} - \)\(26\!\cdots\!44\)\( p^{42} T^{19} + 1667774835599771802 p^{48} T^{20} - 81660828897936 p^{54} T^{21} + 3008908950 p^{60} T^{22} - 76248 p^{66} T^{23} + p^{72} T^{24} \) | |
| 29 | \( 1 + 3348 p T + 6755833788 T^{2} + 12098159708400 p T^{3} + 15402504239709523464 T^{4} + \)\(58\!\cdots\!52\)\( T^{5} + \)\(20\!\cdots\!68\)\( T^{6} + \)\(21\!\cdots\!12\)\( p T^{7} + \)\(18\!\cdots\!72\)\( T^{8} + \)\(17\!\cdots\!28\)\( p T^{9} + \)\(13\!\cdots\!80\)\( T^{10} + \)\(33\!\cdots\!52\)\( T^{11} + \)\(81\!\cdots\!46\)\( T^{12} + \)\(33\!\cdots\!52\)\( p^{6} T^{13} + \)\(13\!\cdots\!80\)\( p^{12} T^{14} + \)\(17\!\cdots\!28\)\( p^{19} T^{15} + \)\(18\!\cdots\!72\)\( p^{24} T^{16} + \)\(21\!\cdots\!12\)\( p^{31} T^{17} + \)\(20\!\cdots\!68\)\( p^{36} T^{18} + \)\(58\!\cdots\!52\)\( p^{42} T^{19} + 15402504239709523464 p^{48} T^{20} + 12098159708400 p^{55} T^{21} + 6755833788 p^{60} T^{22} + 3348 p^{67} T^{23} + p^{72} T^{24} \) | |
| 31 | \( 1 - 21480 T - 3409037526 T^{2} + 60434647346296 T^{3} + 6554148396337073490 T^{4} - \)\(90\!\cdots\!16\)\( T^{5} - \)\(88\!\cdots\!48\)\( T^{6} + \)\(90\!\cdots\!40\)\( T^{7} + \)\(92\!\cdots\!74\)\( T^{8} - \)\(61\!\cdots\!68\)\( T^{9} - \)\(82\!\cdots\!06\)\( T^{10} + \)\(20\!\cdots\!84\)\( T^{11} + \)\(71\!\cdots\!18\)\( T^{12} + \)\(20\!\cdots\!84\)\( p^{6} T^{13} - \)\(82\!\cdots\!06\)\( p^{12} T^{14} - \)\(61\!\cdots\!68\)\( p^{18} T^{15} + \)\(92\!\cdots\!74\)\( p^{24} T^{16} + \)\(90\!\cdots\!40\)\( p^{30} T^{17} - \)\(88\!\cdots\!48\)\( p^{36} T^{18} - \)\(90\!\cdots\!16\)\( p^{42} T^{19} + 6554148396337073490 p^{48} T^{20} + 60434647346296 p^{54} T^{21} - 3409037526 p^{60} T^{22} - 21480 p^{66} T^{23} + p^{72} T^{24} \) | |
| 37 | \( ( 1 + 12768 T + 7314333078 T^{2} - 126861243756448 T^{3} + 23527709642592971919 T^{4} - \)\(92\!\cdots\!84\)\( T^{5} + \)\(63\!\cdots\!12\)\( T^{6} - \)\(92\!\cdots\!84\)\( p^{6} T^{7} + 23527709642592971919 p^{12} T^{8} - 126861243756448 p^{18} T^{9} + 7314333078 p^{24} T^{10} + 12768 p^{30} T^{11} + p^{36} T^{12} )^{2} \) | |
| 41 | \( 1 - 410562 T + 101331354111 T^{2} - 18534534984645606 T^{3} + \)\(27\!\cdots\!59\)\( T^{4} - \)\(34\!\cdots\!44\)\( T^{5} + \)\(38\!\cdots\!40\)\( T^{6} - \)\(37\!\cdots\!20\)\( T^{7} + \)\(32\!\cdots\!37\)\( T^{8} - \)\(26\!\cdots\!74\)\( T^{9} + \)\(20\!\cdots\!73\)\( T^{10} - \)\(15\!\cdots\!94\)\( T^{11} + \)\(10\!\cdots\!38\)\( T^{12} - \)\(15\!\cdots\!94\)\( p^{6} T^{13} + \)\(20\!\cdots\!73\)\( p^{12} T^{14} - \)\(26\!\cdots\!74\)\( p^{18} T^{15} + \)\(32\!\cdots\!37\)\( p^{24} T^{16} - \)\(37\!\cdots\!20\)\( p^{30} T^{17} + \)\(38\!\cdots\!40\)\( p^{36} T^{18} - \)\(34\!\cdots\!44\)\( p^{42} T^{19} + \)\(27\!\cdots\!59\)\( p^{48} T^{20} - 18534534984645606 p^{54} T^{21} + 101331354111 p^{60} T^{22} - 410562 p^{66} T^{23} + p^{72} T^{24} \) | |
| 43 | \( 1 - 71430 T - 13374090363 T^{2} - 1007202949136486 T^{3} + \)\(22\!\cdots\!95\)\( T^{4} + \)\(18\!\cdots\!92\)\( T^{5} - \)\(49\!\cdots\!40\)\( T^{6} - \)\(23\!\cdots\!04\)\( T^{7} - \)\(67\!\cdots\!19\)\( T^{8} + \)\(12\!\cdots\!50\)\( T^{9} + \)\(10\!\cdots\!03\)\( T^{10} - \)\(26\!\cdots\!86\)\( T^{11} - \)\(85\!\cdots\!86\)\( T^{12} - \)\(26\!\cdots\!86\)\( p^{6} T^{13} + \)\(10\!\cdots\!03\)\( p^{12} T^{14} + \)\(12\!\cdots\!50\)\( p^{18} T^{15} - \)\(67\!\cdots\!19\)\( p^{24} T^{16} - \)\(23\!\cdots\!04\)\( p^{30} T^{17} - \)\(49\!\cdots\!40\)\( p^{36} T^{18} + \)\(18\!\cdots\!92\)\( p^{42} T^{19} + \)\(22\!\cdots\!95\)\( p^{48} T^{20} - 1007202949136486 p^{54} T^{21} - 13374090363 p^{60} T^{22} - 71430 p^{66} T^{23} + p^{72} T^{24} \) | |
| 47 | \( 1 + 347652 T + 84414601014 T^{2} + 15340942933575192 T^{3} + \)\(23\!\cdots\!98\)\( T^{4} + \)\(29\!\cdots\!68\)\( T^{5} + \)\(31\!\cdots\!16\)\( T^{6} + \)\(25\!\cdots\!28\)\( T^{7} + \)\(11\!\cdots\!70\)\( T^{8} - \)\(95\!\cdots\!08\)\( T^{9} - \)\(32\!\cdots\!26\)\( T^{10} - \)\(50\!\cdots\!68\)\( T^{11} - \)\(59\!\cdots\!70\)\( T^{12} - \)\(50\!\cdots\!68\)\( p^{6} T^{13} - \)\(32\!\cdots\!26\)\( p^{12} T^{14} - \)\(95\!\cdots\!08\)\( p^{18} T^{15} + \)\(11\!\cdots\!70\)\( p^{24} T^{16} + \)\(25\!\cdots\!28\)\( p^{30} T^{17} + \)\(31\!\cdots\!16\)\( p^{36} T^{18} + \)\(29\!\cdots\!68\)\( p^{42} T^{19} + \)\(23\!\cdots\!98\)\( p^{48} T^{20} + 15340942933575192 p^{54} T^{21} + 84414601014 p^{60} T^{22} + 347652 p^{66} T^{23} + p^{72} T^{24} \) | |
| 53 | \( 1 - 111693428292 T^{2} + \)\(71\!\cdots\!98\)\( T^{4} - \)\(32\!\cdots\!48\)\( T^{6} + \)\(11\!\cdots\!83\)\( T^{8} - \)\(33\!\cdots\!32\)\( T^{10} + \)\(81\!\cdots\!92\)\( T^{12} - \)\(33\!\cdots\!32\)\( p^{12} T^{14} + \)\(11\!\cdots\!83\)\( p^{24} T^{16} - \)\(32\!\cdots\!48\)\( p^{36} T^{18} + \)\(71\!\cdots\!98\)\( p^{48} T^{20} - 111693428292 p^{60} T^{22} + p^{72} T^{24} \) | |
| 59 | \( 1 + 369738 T + 217351680405 T^{2} + 63514684683965466 T^{3} + \)\(23\!\cdots\!83\)\( T^{4} + \)\(59\!\cdots\!24\)\( T^{5} + \)\(16\!\cdots\!96\)\( T^{6} + \)\(35\!\cdots\!84\)\( T^{7} + \)\(77\!\cdots\!93\)\( T^{8} + \)\(16\!\cdots\!86\)\( T^{9} + \)\(28\!\cdots\!55\)\( T^{10} + \)\(63\!\cdots\!38\)\( T^{11} + \)\(11\!\cdots\!62\)\( T^{12} + \)\(63\!\cdots\!38\)\( p^{6} T^{13} + \)\(28\!\cdots\!55\)\( p^{12} T^{14} + \)\(16\!\cdots\!86\)\( p^{18} T^{15} + \)\(77\!\cdots\!93\)\( p^{24} T^{16} + \)\(35\!\cdots\!84\)\( p^{30} T^{17} + \)\(16\!\cdots\!96\)\( p^{36} T^{18} + \)\(59\!\cdots\!24\)\( p^{42} T^{19} + \)\(23\!\cdots\!83\)\( p^{48} T^{20} + 63514684683965466 p^{54} T^{21} + 217351680405 p^{60} T^{22} + 369738 p^{66} T^{23} + p^{72} T^{24} \) | |
| 61 | \( 1 - 135744 T - 246761883948 T^{2} + 30073967562178432 T^{3} + \)\(34\!\cdots\!08\)\( T^{4} - \)\(36\!\cdots\!72\)\( T^{5} - \)\(34\!\cdots\!48\)\( T^{6} + \)\(28\!\cdots\!52\)\( T^{7} + \)\(26\!\cdots\!76\)\( T^{8} - \)\(14\!\cdots\!64\)\( T^{9} - \)\(16\!\cdots\!00\)\( T^{10} + \)\(30\!\cdots\!80\)\( T^{11} + \)\(93\!\cdots\!50\)\( T^{12} + \)\(30\!\cdots\!80\)\( p^{6} T^{13} - \)\(16\!\cdots\!00\)\( p^{12} T^{14} - \)\(14\!\cdots\!64\)\( p^{18} T^{15} + \)\(26\!\cdots\!76\)\( p^{24} T^{16} + \)\(28\!\cdots\!52\)\( p^{30} T^{17} - \)\(34\!\cdots\!48\)\( p^{36} T^{18} - \)\(36\!\cdots\!72\)\( p^{42} T^{19} + \)\(34\!\cdots\!08\)\( p^{48} T^{20} + 30073967562178432 p^{54} T^{21} - 246761883948 p^{60} T^{22} - 135744 p^{66} T^{23} + p^{72} T^{24} \) | |
| 67 | \( 1 + 289938 T - 239694526299 T^{2} - 44192063312631998 T^{3} + \)\(30\!\cdots\!39\)\( T^{4} + \)\(52\!\cdots\!36\)\( T^{5} - \)\(38\!\cdots\!24\)\( T^{6} + \)\(35\!\cdots\!96\)\( T^{7} + \)\(47\!\cdots\!53\)\( T^{8} - \)\(12\!\cdots\!98\)\( T^{9} - \)\(34\!\cdots\!89\)\( T^{10} - \)\(29\!\cdots\!22\)\( T^{11} + \)\(22\!\cdots\!50\)\( T^{12} - \)\(29\!\cdots\!22\)\( p^{6} T^{13} - \)\(34\!\cdots\!89\)\( p^{12} T^{14} - \)\(12\!\cdots\!98\)\( p^{18} T^{15} + \)\(47\!\cdots\!53\)\( p^{24} T^{16} + \)\(35\!\cdots\!96\)\( p^{30} T^{17} - \)\(38\!\cdots\!24\)\( p^{36} T^{18} + \)\(52\!\cdots\!36\)\( p^{42} T^{19} + \)\(30\!\cdots\!39\)\( p^{48} T^{20} - 44192063312631998 p^{54} T^{21} - 239694526299 p^{60} T^{22} + 289938 p^{66} T^{23} + p^{72} T^{24} \) | |
| 71 | \( 1 - 791865391260 T^{2} + \)\(34\!\cdots\!22\)\( T^{4} - \)\(10\!\cdots\!48\)\( T^{6} + \)\(22\!\cdots\!99\)\( T^{8} - \)\(39\!\cdots\!04\)\( T^{10} + \)\(56\!\cdots\!96\)\( T^{12} - \)\(39\!\cdots\!04\)\( p^{12} T^{14} + \)\(22\!\cdots\!99\)\( p^{24} T^{16} - \)\(10\!\cdots\!48\)\( p^{36} T^{18} + \)\(34\!\cdots\!22\)\( p^{48} T^{20} - 791865391260 p^{60} T^{22} + p^{72} T^{24} \) | |
| 73 | \( ( 1 + 488850 T + 619237602795 T^{2} + 262395918353767622 T^{3} + \)\(19\!\cdots\!39\)\( T^{4} + \)\(68\!\cdots\!92\)\( T^{5} + \)\(36\!\cdots\!30\)\( T^{6} + \)\(68\!\cdots\!92\)\( p^{6} T^{7} + \)\(19\!\cdots\!39\)\( p^{12} T^{8} + 262395918353767622 p^{18} T^{9} + 619237602795 p^{24} T^{10} + 488850 p^{30} T^{11} + p^{36} T^{12} )^{2} \) | |
| 79 | \( 1 + 764796 T - 496629231330 T^{2} - 638169693294513176 T^{3} + \)\(42\!\cdots\!86\)\( T^{4} + \)\(24\!\cdots\!16\)\( T^{5} + \)\(27\!\cdots\!00\)\( T^{6} - \)\(67\!\cdots\!56\)\( T^{7} - \)\(17\!\cdots\!58\)\( T^{8} + \)\(14\!\cdots\!36\)\( T^{9} + \)\(75\!\cdots\!10\)\( T^{10} - \)\(16\!\cdots\!16\)\( T^{11} - \)\(22\!\cdots\!34\)\( T^{12} - \)\(16\!\cdots\!16\)\( p^{6} T^{13} + \)\(75\!\cdots\!10\)\( p^{12} T^{14} + \)\(14\!\cdots\!36\)\( p^{18} T^{15} - \)\(17\!\cdots\!58\)\( p^{24} T^{16} - \)\(67\!\cdots\!56\)\( p^{30} T^{17} + \)\(27\!\cdots\!00\)\( p^{36} T^{18} + \)\(24\!\cdots\!16\)\( p^{42} T^{19} + \)\(42\!\cdots\!86\)\( p^{48} T^{20} - 638169693294513176 p^{54} T^{21} - 496629231330 p^{60} T^{22} + 764796 p^{66} T^{23} + p^{72} T^{24} \) | |
| 83 | \( 1 + 396900 T + 1073338925046 T^{2} + 405167051947757400 T^{3} + \)\(64\!\cdots\!22\)\( T^{4} + \)\(20\!\cdots\!36\)\( T^{5} + \)\(19\!\cdots\!68\)\( T^{6} + \)\(25\!\cdots\!44\)\( T^{7} + \)\(82\!\cdots\!42\)\( T^{8} - \)\(25\!\cdots\!60\)\( T^{9} - \)\(24\!\cdots\!74\)\( T^{10} - \)\(18\!\cdots\!80\)\( T^{11} - \)\(11\!\cdots\!18\)\( T^{12} - \)\(18\!\cdots\!80\)\( p^{6} T^{13} - \)\(24\!\cdots\!74\)\( p^{12} T^{14} - \)\(25\!\cdots\!60\)\( p^{18} T^{15} + \)\(82\!\cdots\!42\)\( p^{24} T^{16} + \)\(25\!\cdots\!44\)\( p^{30} T^{17} + \)\(19\!\cdots\!68\)\( p^{36} T^{18} + \)\(20\!\cdots\!36\)\( p^{42} T^{19} + \)\(64\!\cdots\!22\)\( p^{48} T^{20} + 405167051947757400 p^{54} T^{21} + 1073338925046 p^{60} T^{22} + 396900 p^{66} T^{23} + p^{72} T^{24} \) | |
| 89 | \( 1 - 4645466908068 T^{2} + \)\(10\!\cdots\!26\)\( T^{4} - \)\(14\!\cdots\!96\)\( T^{6} + \)\(14\!\cdots\!55\)\( T^{8} - \)\(10\!\cdots\!08\)\( T^{10} + \)\(60\!\cdots\!00\)\( T^{12} - \)\(10\!\cdots\!08\)\( p^{12} T^{14} + \)\(14\!\cdots\!55\)\( p^{24} T^{16} - \)\(14\!\cdots\!96\)\( p^{36} T^{18} + \)\(10\!\cdots\!26\)\( p^{48} T^{20} - 4645466908068 p^{60} T^{22} + p^{72} T^{24} \) | |
| 97 | \( 1 + 38874 T - 3214389989361 T^{2} + 836649759038799430 T^{3} + \)\(54\!\cdots\!75\)\( T^{4} - \)\(24\!\cdots\!92\)\( T^{5} - \)\(59\!\cdots\!00\)\( T^{6} + \)\(35\!\cdots\!64\)\( T^{7} + \)\(47\!\cdots\!17\)\( T^{8} - \)\(28\!\cdots\!70\)\( T^{9} - \)\(30\!\cdots\!91\)\( T^{10} + \)\(10\!\cdots\!10\)\( T^{11} + \)\(21\!\cdots\!58\)\( T^{12} + \)\(10\!\cdots\!10\)\( p^{6} T^{13} - \)\(30\!\cdots\!91\)\( p^{12} T^{14} - \)\(28\!\cdots\!70\)\( p^{18} T^{15} + \)\(47\!\cdots\!17\)\( p^{24} T^{16} + \)\(35\!\cdots\!64\)\( p^{30} T^{17} - \)\(59\!\cdots\!00\)\( p^{36} T^{18} - \)\(24\!\cdots\!92\)\( p^{42} T^{19} + \)\(54\!\cdots\!75\)\( p^{48} T^{20} + 836649759038799430 p^{54} T^{21} - 3214389989361 p^{60} T^{22} + 38874 p^{66} T^{23} + p^{72} T^{24} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.30529418855622910169977211532, −4.25741531740710876721639646924, −3.98833667757479110446945930866, −3.82380658778040611740421074081, −3.81835754126638731571345322305, −3.81386767066920974307651180733, −3.22540061671477517766861859061, −3.00744885810876573145583567521, −2.98890384039902888041420914703, −2.98588262674021069536167170950, −2.97564220368788261497021050132, −2.97269623851501650507693244289, −2.38534056675353068575268222684, −2.17494291539105406321199994127, −2.17032737102348169732812701438, −1.97441739809685778773307534652, −1.57775618834157006350514906434, −1.36193983612611730777324212533, −1.34601394346812884647110319282, −1.11436100115256290539758223006, −0.870856977876909844757969776172, −0.820851924587811420712100752356, −0.54918675981074221086355037558, −0.083237511094538350223582725239, −0.04200645038317786333095708969, 0.04200645038317786333095708969, 0.083237511094538350223582725239, 0.54918675981074221086355037558, 0.820851924587811420712100752356, 0.870856977876909844757969776172, 1.11436100115256290539758223006, 1.34601394346812884647110319282, 1.36193983612611730777324212533, 1.57775618834157006350514906434, 1.97441739809685778773307534652, 2.17032737102348169732812701438, 2.17494291539105406321199994127, 2.38534056675353068575268222684, 2.97269623851501650507693244289, 2.97564220368788261497021050132, 2.98588262674021069536167170950, 2.98890384039902888041420914703, 3.00744885810876573145583567521, 3.22540061671477517766861859061, 3.81386767066920974307651180733, 3.81835754126638731571345322305, 3.82380658778040611740421074081, 3.98833667757479110446945930866, 4.25741531740710876721639646924, 4.30529418855622910169977211532
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.